Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 41 / 2 / Solution

Let $m={\overline{Y}}_{+++}$, $r_i={\overline{Y}}_{i++}$ and $c_j={\overline{Y}}_{+j+}$. The proposed least-squares estimators satisfy the sum-to-zero constraints because ${\sum }_i{r}_i=Im$ and ${\sum }_j{c}_j=Jm$. Define $e_{ijk}=Y_{ijk}-r_i-c_j+m$. Direct summation givesNow write any other admissible parameters as $\mu =m+u$, ${\alpha }_i=r_i-m+a_i$, ${\beta }_j=c_j-m+b_j$, where ${\sum }_i{a}_i={\sum }_j{b}_j=0$. Expanding the ordinary least squares criterion, the cross-products with $e$ vanish by the preceding identities. The cross-products between $u$, $a_i$ and $b_j$ vanish by their zero sums. ConsequentlyEvery added term is nonnegative and all vanish only at $u=0$, $a_i=b_j=0$. This proves the unique constrained minimum, withThe additive-model residual sum of squares and intercept-only residual sum of squares are thereforeThe same orthogonality gives the analysis of variance identityFitting factor B alone yields fitted values $c_j$ and residual sum ${\mathrm{RSS}}_0-S{S}_B$. Fitting A alone gives residual sum ${\mathrm{RSS}}_0-S{S}_A$. Adding B after A then reduces it to ${\mathrm{RSS}}_1$, a reduction of $S{S}_B$ again. The reduction due to B is $S{S}_B$ in either order. This is balanced factorial orthogonality; it generally fails for unequal cell replication.

There are $12$ observations. The additive model estimates $1+(3-1)+(2-1)=4$ independent parameters, so the missing degrees of freedom are A: 2, B: 1, residuals: 8. The individual requested residual sums are below.

To check for an interaction, extend the normal linear model to $\mu +{\alpha }_i+{\beta }_j+{\gamma }_{ij}$, with each row and column sum of $\gamma$ zero. Equivalently, fit the six unrestricted cell means. Its residual sum of squares is the within-cell sumThe interaction adds $(I-1)(J-1)=2$ parameters; the cell-means residual has $IJ(K-1)=6$ degrees of freedom. Under the no-interaction null hypothesis and the independent equal-variance normal error model,The numerator and denominator arise from orthogonal Gaussian projections, proving the exact F-test. Reject for a sufficiently large value. The printed additive analysis of variance table does not supply ${\mathrm{RSS}}_{\mathrm{cell}}$, so it cannot determine this test statistic numerically; the replicated cell data would do so.